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<br> <h1> Link to the source code for this example: </h1>
<a href="https://github.com/libMesh/libmesh/tree/master/examples/adjoints/adjoints_ex4" target="_blank">Open adjoints_ex4 in new tab.</a>
<a name="output"></a> 
<br><br><br> <h1> The console output of the program: </h1> 
<pre>
***************************************************************
* Running Example adjoints_ex4:
*   example-opt  
***************************************************************
 
Started /Users/petejw/projects/libmesh_git/build/examples/adjoints/adjoints_ex4/.libs/example-opt
Reading in and building the mesh
Building system
Initializing systems
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=65
    n_local_nodes()=65
  n_elem()=15
    n_local_elem()=15
    n_active_elem()=12
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "LaplaceSystem"
    Type "Implicit"
    Variables="T" 
    Finite Element Types="LAGRANGE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="SECOND", "THIRD" 
    n_dofs()=65
    n_local_dofs()=65
    n_constrained_dofs()=0
    n_local_constrained_dofs()=0
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 12.8154
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 25
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 0

  Nonlinear solver converged, step 0, residual reduction 9.41161e-11 < 1e-09
Adaptive step 0, we have 12 active elements and 65 active dofs.
Postprocessing: 
The relative error in QoI 0 is 1.7448570691601495
The relative error in QoI 1 is 0.0044573232594091149

Computing the error estimate using the Adjoint Refinement Error Estimator

The computed relative error in QoI 0 is 0.00019007492686886688
The computed relative error in QoI 1 is 0.0045714923983656815

The effectivity index for the computed error in QoI 0 is 0.00010893438220722313
The effectivity index for the computed error in QoI 1 is 1.0256138342031988

Refined mesh to 48 active elements and 225 active dofs.
  Nonlinear solver converged, step 0, residual reduction 5.8027048027426932e-11 < 1.0000000000000001e-09
Adaptive step 1, we have 48 active elements and 225 active dofs.
Postprocessing: 
The relative error in QoI 0 is 0.0002799284426589448
The relative error in QoI 1 is 0.00047776972716113617

Computing the error estimate using the Adjoint Refinement Error Estimator

The computed relative error in QoI 0 is 0.00022987273088688831
The computed relative error in QoI 1 is 0.0005940412659918181

The effectivity index for the computed error in QoI 0 is 0.82118390222660342
The effectivity index for the computed error in QoI 1 is 1.24336313546184

Refined mesh to 192 active elements and 833 active dofs.
  Nonlinear solver converged, step 0, residual reduction 5.6736077195905133e-11 < 1.0000000000000001e-09
Adaptive step 2, we have 192 active elements and 833 active dofs.
Postprocessing: 
The relative error in QoI 0 is 0.00012462746003517845
The relative error in QoI 1 is 0.00012048195197492737

Computing the error estimate using the Adjoint Refinement Error Estimator

The computed relative error in QoI 0 is 0.00010470074642364172
The computed relative error in QoI 1 is 5.8186069998719015e-05

The effectivity index for the computed error in QoI 0 is 0.84010976709376861
The effectivity index for the computed error in QoI 1 is 0.48294428372830228


 -------------------------------------------------------------------------------------------------------------------
| Time:           Wed Apr 13 10:22:07 2016                                                                          |
| OS:             Darwin                                                                                            |
| HostName:       peterson-laptop.local                                                                             |
| OS Release:     15.0.0                                                                                            |
| OS Version:     Darwin Kernel Version 15.0.0: Sat Sep 19 15:53:46 PDT 2015; root:xnu-3247.10.11~1/RELEASE_X86_64  |
| Machine:        x86_64                                                                                            |
| Username:       petejw                                                                                            |
| Configuration:  ../configure  'INSTALL=/usr/bin/install -C'                                                       |
|  '--with-methods=opt dbg'                                                                                         |
|  '--prefix=/Users/petejw/projects/libmesh_git/installed'                                                          |
|  '--enable-perflog'                                                                                               |
|  '--enable-tracefiles'                                                                                            |
|  '--disable-blocked-storage'                                                                                      |
|  '--enable-default-comm-world'                                                                                    |
|  '--enable-tecio'                                                                                                 |
|  '--enable-unique-ptr'                                                                                            |
|  '--enable-ifem'                                                                                                  |
|  'PETSC_DIR=/opt/moose/petsc/mpich_petsc-3.6.1/clang-opt-superlu'                                                 |
|  'SLEPC_DIR=/opt/petejw/slepc/3.6.1'                                                                              |
|  'CPPFLAGS= '                                                                                                     |
|  'LIBS= '                                                                                                         |
|  'CPPFLAGS=  '                                                                                                    |
|  'LIBS=  '                                                                                                        |
|  '--disable-netcdf-4'                                                                                             |
|  '--disable-testsets'                                                                                             |
|  'CXX=mpicxx'                                                                                                     |
|  'CC=mpicc'                                                                                                       |
|  'F77=mpif77'                                                                                                     |
|  'FC=mpif90'                                                                                                      |
|  'CPPFLAGS=   '                                                                                                   |
|  'LIBS=   '                                                                                                       |
 -------------------------------------------------------------------------------------------------------------------
 -----------------------------------------------------------------------------------------------------------------
| libMesh Performance: Alive time=3.76453, Active time=3.2869                                                     |
 -----------------------------------------------------------------------------------------------------------------
| Event                              nCalls     Total Time  Avg Time    Total Time  Avg Time    % of Active Time  |
|                                               w/o Sub     w/o Sub     With Sub    With Sub    w/o S    With S   |
|-----------------------------------------------------------------------------------------------------------------|
|                                                                                                                 |
|                                                                                                                 |
| DofMap                                                                                                          |
|   add_neighbors_to_send_list()     15         0.0018      0.000122    0.0018      0.000122    0.06     0.06     |
|   build_sparsity()                 15         0.0138      0.000917    0.0158      0.001057    0.42     0.48     |
|   create_dof_constraints()         15         0.0017      0.000111    0.0017      0.000111    0.05     0.05     |
|   distribute_dofs()                15         0.0033      0.000219    0.0194      0.001294    0.10     0.59     |
|   dof_indices()                    78804      0.0195      0.000000    0.0195      0.000000    0.59     0.59     |
|   old_dof_indices()                57600      0.0185      0.000000    0.0185      0.000000    0.56     0.56     |
|   prepare_send_list()              16         0.0000      0.000000    0.0000      0.000000    0.00     0.00     |
|   reinit()                         15         0.0161      0.001074    0.0161      0.001074    0.49     0.49     |
|                                                                                                                 |
| EquationSystems                                                                                                 |
|   build_solution_vector()          9          0.0013      0.000140    0.0016      0.000179    0.04     0.05     |
|                                                                                                                 |
| FE                                                                                                              |
|   compute_shape_functions()        94460      0.0808      0.000001    0.0808      0.000001    2.46     2.46     |
|   init_shape_functions()           81128      0.6978      0.000009    0.6978      0.000009    21.23    21.23    |
|   inverse_map()                    289392     0.1966      0.000001    0.1966      0.000001    5.98     5.98     |
|                                                                                                                 |
| FEMSystem                                                                                                       |
|   assemble_qoi_derivative()        6          0.0113      0.001883    0.0314      0.005239    0.34     0.96     |
|   assembly()                       3          0.0015      0.000498    0.0067      0.002221    0.05     0.20     |
|   assembly(get_jacobian)           6          0.0141      0.002346    0.0721      0.012022    0.43     2.19     |
|   assembly(get_residual)           6          0.0101      0.001677    0.0327      0.005446    0.31     0.99     |
|   numerical_elem_jacobian()        4536       0.0367      0.000008    0.0367      0.000008    1.12     1.12     |
|   numerical_side_jacobian()        672        0.0037      0.000006    0.0037      0.000006    0.11     0.11     |
|   postprocess()                    3          0.0009      0.000298    0.0033      0.001100    0.03     0.10     |
|                                                                                                                 |
| FEMap                                                                                                           |
|   compute_affine_map()             94460      0.1054      0.000001    0.1054      0.000001    3.21     3.21     |
|   compute_face_map()               1904       0.0031      0.000002    0.0077      0.000004    0.09     0.23     |
|   init_face_shape_functions()      24         0.0000      0.000002    0.0000      0.000002    0.00     0.00     |
|   init_reference_to_physical_map() 81128      0.2855      0.000004    0.2855      0.000004    8.69     8.69     |
|                                                                                                                 |
| GMVIO                                                                                                           |
|   write_nodal_data()               9          0.0135      0.001495    0.0135      0.001495    0.41     0.41     |
|                                                                                                                 |
| GenericProjector                                                                                                |
|   operator()                       36         0.0369      0.001026    2.7065      0.075181    1.12     82.34    |
|   project_edges                    18360      0.0011      0.000000    0.0011      0.000000    0.03     0.03     |
|   project_interior                 18360      0.0018      0.000000    0.0018      0.000000    0.05     0.05     |
|   project_nodes                    18360      0.0552      0.000003    2.6474      0.000144    1.68     80.54    |
|   project_sides                    18360      0.0011      0.000000    0.0011      0.000000    0.03     0.03     |
|                                                                                                                 |
| ImplicitSystem                                                                                                  |
|   adjoint_solve()                  6          0.0001      0.000018    0.3377      0.056283    0.00     10.27    |
|                                                                                                                 |
| Mesh                                                                                                            |
|   all_second_order()               1          0.0000      0.000037    0.0000      0.000037    0.00     0.00     |
|   contract()                       14         0.0034      0.000241    0.0139      0.000993    0.10     0.42     |
|   find_neighbors()                 17         0.0260      0.001530    0.0260      0.001530    0.79     0.79     |
|   renumber_nodes_and_elem()        14         0.0105      0.000751    0.0105      0.000751    0.32     0.32     |
|                                                                                                                 |
| MeshOutput                                                                                                      |
|   write_equation_systems()         9          0.0001      0.000010    0.0152      0.001684    0.00     0.46     |
|                                                                                                                 |
| MeshRefinement                                                                                                  |
|   _coarsen_elements()              20         0.0023      0.000113    0.0023      0.000113    0.07     0.07     |
|   _refine_elements()               23         0.0156      0.000677    0.0317      0.001379    0.47     0.97     |
|   add_node()                       47628      0.0121      0.000000    0.0121      0.000000    0.37     0.37     |
|   make_coarsening_compatible()     14         0.0036      0.000258    0.0036      0.000258    0.11     0.11     |
|   make_flags_parallel_consistent() 28         0.0012      0.000044    0.0012      0.000044    0.04     0.04     |
|   make_refinement_compatible()     14         0.0002      0.000011    0.0002      0.000011    0.00     0.00     |
|                                                                                                                 |
| NewtonSolver                                                                                                    |
|   solve()                          3          0.0002      0.000077    0.0131      0.004377    0.01     0.40     |
|                                                                                                                 |
| OldSolutionValue                                                                                                |
|   Number eval_at_node()            165240     0.0115      0.000000    0.0115      0.000000    0.35     0.35     |
|   check_old_context(c,p)           79200      0.0416      0.000001    0.1035      0.000001    1.26     3.15     |
|   eval_at_point()                  79200      1.2439      0.000016    2.5609      0.000032    37.85    77.91    |
|                                                                                                                 |
| Parallel                                                                                                        |
|   allgather()                      15         0.0000      0.000000    0.0000      0.000000    0.00     0.00     |
|                                                                                                                 |
| Partitioner                                                                                                     |
|   set_node_processor_ids()         12         0.0021      0.000176    0.0021      0.000176    0.06     0.06     |
|   single_partition()               5          0.0000      0.000007    0.0000      0.000007    0.00     0.00     |
|                                                                                                                 |
| PetscLinearSolver                                                                                               |
|   solve()                          15         0.2370      0.015799    0.2370      0.015799    7.21     7.21     |
|                                                                                                                 |
| System                                                                                                          |
|   project_vector(old,new)          36         0.0180      0.000500    2.7350      0.075972    0.55     83.21    |
|                                                                                                                 |
| TopologyMap                                                                                                     |
|   init()                           23         0.0246      0.001071    0.0246      0.001071    0.75     0.75     |
 -----------------------------------------------------------------------------------------------------------------
| Totals:                            1229254    3.2869                                          100.00            |
 -----------------------------------------------------------------------------------------------------------------

 
***************************************************************
* Done Running Example adjoints_ex4:
*   example-opt  
***************************************************************
</pre>
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